Mod $p$ Iwasawa algebras of pro-$p$ Iwahori subgroups

Abstract: Suppose $F$ is a finite unramified extension of $\mathbb{Q}p$, and $G$ is the group of $F$-points of a split, connected, reductive group over $F$. Under a natural restriction on $p$, we determine the structure of the graded mod $p$ Iwasawa algebra $\textrm{gr}{\mathfrak{m}}(\mathbb{F}_p [![ I]!])$, where $I$ is a pro-$p$ Iwahori subgroup of $G$. We also determine its maximal commutative quotient, and relate these results to Gelfand--Kirillov dimensions of smooth mod $p$ representations of $G$.